Why a Diagnostic Checklist Is Needed
An associated single-decrement table describes the probability of exit from one specified cause when the other competing causes are removed under a stated assumption. Its probabilities are not, in general, the same as the cause-specific exit probabilities observed when all causes operate together. A table can therefore contain individually plausible probabilities and still be inconsistent with its stated basis or with its reported multiple-decrement exits.
A reliable audit follows a fixed order: identify what each column represents, identify the basis, check probability bounds, calculate survival, and reconstruct the cause-specific exits. The calculations below use the uniform distribution of decrements (UDD) basis unless stated otherwise. All product and table figures are hypothetical.
The Checklist: Meaning, Basis, Bounds, Survival, Exits
Identify whether a probability is associated single-decrement \(q'_j\) or multiple-decrement \(q_j\). Confirm that the values refer to the same age, duration, and one-year period, and that each is conditional on being in force at the start of that period.
Record whether the conversion uses UDD, constant force, or another stated assumption. A reconstruction made under one basis cannot be used to validate figures calculated under another.
Each probability must be between 0 and 1. For an associated table, each \(q'_j\) must satisfy \(0\leq q'_j\leq1\). Also check that any reported multiple-decrement probabilities are nonnegative and that their sum cannot exceed 1.
Under independent associated single-decrement UDD distributions, survival through the year with every cause operating is the product of the cause-specific associated survivals. Do not subtract the sum of associated probabilities from 1.
Under the stated basis, calculate the multiple-decrement probability for each cause. Check that the reconstructed exits and survival sum to 1, and multiply by the starting number of lives when auditing counts.
For two causes under UDD, let \(t\) be the fraction of the year elapsed. In the associated single-decrement model, the probability of still being free of cause \(j\) at time \(t\) is \(1-tq'_j\). With independent associated decrement times, the probability of still being free of both causes is the product of these two survivals.
The density of exit from cause 1 at time \(t\), while also remaining free of cause 2 until that time, is \(q'_1(1-tq'_2)\). Integrating this density over the year gives the multiple-decrement probability for cause 1. Reversing the cause labels gives the corresponding expression for cause 2.
This derivation explains why the associated probability for cause 1 is not itself the observed multiple-decrement probability for cause 1: some lives leave through cause 2 before cause 1 can occur. The reconstructed exits must add to \(1-p^{\text{all}}\). For more than two causes under the same independent UDD construction, the survival is the product of all associated survivals, and the cause-\(j\) exit probability is the integral of \(q'_j\) multiplied by the other causes’ survivals through time \(t\).
Worked Audits
Worked Example: Reconstructing Two Cause Exits
A hypothetical table for 1,000 active policies gives associated one-year UDD probabilities \(q'_1=0.20\) for disability and \(q'_2=0.10\) for death. The accompanying multiple-decrement column reports 190 disability exits and 90 death exits. Check the probabilities and counts.
Method. Apply the two-cause UDD reconstruction. First, both associated probabilities are within the required bounds: \(0\leq0.20\leq1\) and \(0\leq0.10\leq1\).
The reconstructed multiple-decrement probability for disability is \(q_1=0.20(1-0.10/2)=0.20(0.95)=0.19\). For death it is \(q_2=0.10(1-0.20/2)=0.10(0.90)=0.09\).
The probability of surviving the year with neither decrement is \(p^{\text{all}}=(1-0.20)(1-0.10)=0.80(0.90)=0.72\). The probability check is \(0.19+0.09+0.72=1.00\).
Multiplying by 1,000 starting policies gives 190 disability exits, 90 death exits, and \(1{,}000(0.72)=720\) survivors. These counts sum to \(190+90+720=1{,}000\), so the table reconciles.
Common mistake avoided: Using 200 and 100 as the multiple-decrement exit counts confuses associated probabilities with competing-risk exits. The other cause reduces each cause’s reconstructed exit probability.
Worked Example: Finding a Cause-Exit Discrepancy
A hypothetical group of 800 contracts has associated probabilities \(q'_1=0.25\) and \(q'_2=0.20\) on a UDD basis. A report states that there were 180 exits from cause 1, 160 from cause 2, and 460 survivors. Audit the report.
Method. Check the associated probabilities first: both lie in \([0,1]\). Reconstruct the multiple-decrement probabilities under the stated two-cause UDD basis: \(q_1=0.25(1-0.20/2)=0.25(0.90)=0.225\), and \(q_2=0.20(1-0.25/2)=0.20(0.875)=0.175\).
The all-cause survival probability is \(p^{\text{all}}=(1-0.25)(1-0.20)=0.75(0.80)=0.60\). Thus the expected counts are \(800(0.225)=180\) exits from cause 1, \(800(0.175)=140\) exits from cause 2, and \(800(0.60)=480\) survivors.
The reported cause-2 exit count is 160 rather than 140, and the reported survivor count is 460 rather than 480. The reported figures do sum to 800, but that total alone does not make them consistent with the associated table: the separate cause counts fail the UDD reconstruction. The reconstructed figures check as \(180+140+480=800\).
Common mistake avoided: Checking only that the reported counts add to the starting population. A valid audit must also confirm that each cause count agrees with the stated associated probabilities and basis.
Worked Example: Detecting a Basis Mismatch
A hypothetical table has \(q'_1=0.20\) and \(q'_2=0.10\). One analyst uses UDD; another treats the associated decrements as independent constant-force risks. Calculate the implied multiple-decrement exits under each interpretation, and explain why the results should not be mixed.
Method under UDD. From the two-cause UDD calculations, \(q_1=0.20(0.95)=0.1900\), \(q_2=0.10(0.90)=0.0900\), and survival is \(0.80(0.90)=0.7200\). The probabilities sum to 1.
Method under constant force. The cause-specific forces corresponding to the associated single-decrement probabilities are \(\mu_1=-\ln(1-0.20)=-\ln(0.80)=0.2231436\) and \(\mu_2=-\ln(1-0.10)=-\ln(0.90)=0.1053605\). The total force is \(\mu=0.2231436+0.1053605=0.3285041\). Total survival is \(e^{-\mu}=0.80(0.90)=0.7200\), so total exit probability is \(1-0.7200=0.2800\).
Under constant force, the cause-specific exit probabilities are the cause’s share of the total force multiplied by the total exit probability: \(q_1=(0.2231436/0.3285041)(0.2800)\approx0.1902\), and \(q_2=(0.1053605/0.3285041)(0.2800)\approx0.0898\). Rounded to four decimal places, the constant-force exits and survival are \(0.1902\), \(0.0898\), and \(0.7200\); they sum to 1.0000.
The UDD and constant-force reconstructions differ slightly even though the associated probabilities are the same. The figures are close here, but that does not justify mixing them in one table. A table based on UDD should be audited with UDD reconstruction throughout.
Common mistake avoided: Assuming a matching total survival proves that cause-specific exits match. The total survival happens to agree under these two independent associated models, but the cause allocation depends on the basis.
What the Checks Establish—and What They Do Not
The checklist can identify a mislabeled probability, an invalid value, an incorrect survival calculation, or a cause exit that fails reconstruction. A successful numerical reconciliation shows internal consistency under the stated assumptions; it does not establish that the chosen decrement basis is appropriate for a particular product or population. The basis must come from the model being used, not from whichever calculation produces a preferred result.
Keep the following distinction explicit when reporting an audit:
| Quantity | Meaning | Two-cause UDD relation |
|---|---|---|
| \(q'_j\) | Associated single-decrement probability for cause \(j\), with other causes removed | Given by the associated table |
| \(q_j\) | Probability of exit from cause \(j\) while both causes operate | \(q_j=q'_j(1-q'_k/2)\), where \(k\ne j\) |
| \(p^{\text{all}}\) | Probability of surviving the year with neither decrement | \((1-q'_1)(1-q'_2)\) |
A useful final check is to state the three components together: the assumed basis, the reconstructed cause-specific exits, and all-cause survival. For a two-cause UDD calculation, the probability identity is \(q_1+q_2+p^{\text{all}}=1\). For a table of \(l\) starting lives, the corresponding count check is that the cause-specific exits plus survivors equal \(l\), subject to the table’s rounding convention. When those checks fail, return to the labels and basis before changing any values.
Check Your Understanding
For each problem, apply the stated basis consistently and check the reconstructed probabilities or counts.
- Under two-cause UDD, an associated table gives \(q'_1=0.12\) and \(q'_2=0.08\). Find both multiple-decrement exit probabilities and the all-cause survival probability. (Answer: \(q_1=0.1152,\ q_2=0.0752,\ p^{\text{all}}=0.8096\).)
- A hypothetical group has 2,500 lives and associated UDD probabilities \(q'_1=0.16\), \(q'_2=0.10\). Find the reconstructed exit counts for each cause and the survivor count. (Answer: 380 exits from cause 1, 230 exits from cause 2, and 1,890 survivors.)
- A table for 1,200 lives gives \(q'_1=0.30\), \(q'_2=0.20\) under UDD, but reports 330 cause-1 exits, 210 cause-2 exits, and 660 survivors. Reconstruct the consistent counts and identify whether the reported figures pass. (Answer: The consistent counts are 324, 204, and 672; the reported figures fail reconstruction.)
- For associated probabilities \(q'_1=0.20\) and \(q'_2=0.10\), state the two UDD multiple-decrement exit probabilities and the two constant-force exit probabilities, each to four decimal places. (Answer: UDD: \(0.1900,\ 0.0900\); constant force: \(0.1902,\ 0.0898\).)
- Three independent associated UDD decrements have probabilities \(q'_1=0.10\), \(q'_2=0.20\), and \(q'_3=0.30\). Find the all-cause survival and the multiple-decrement exit probability for each cause. (Answer: Survival \(=0.504\); exits \(q_1=0.077,\ q_2=0.162,\ q_3=0.257\), which sum with survival to 1.)